game of thrones - Assume R+L=J is true. How can that ever be proven in the eyes of Westeros?

So let's assume that the relatively ambiguous Tower of Joy scene in season 6 confirms that Jon is indeed

Lyanna Stark's child, sired by Rhaegar Targaryan.

In that case, considering the current situation in Westeros as of season 7 (or as of book 5 + released chapters of book 6), how does this get proven in the eyes of the lords, ladies, and commoners of Westeros?

It feels like an impossible claim to prove given that Eddard Stark is dead, Howland Reed seems to have vanished off the face of Planetos, and anyone else who was alive at the time with that knowledge appears to be dead and gone. EVEN IF Howland Reed manages to crawl out of the woodwork to corroborate Bran/3-eyed Raven's claim (assuming he even makes that claim) about Jon's true heritage, we're still just talking about the word of two relatively unknown (outside of the North) Westerosi noblemen.

The only way I can see of "proving" anything is if Jon ends up riding a dragon. But even then, would that, in addition to the theoretical claims of Howland Reed and Bran Stark be enough to prove that Jon Snow is actually Jon Targaryan (assuming R and L actually got married, and assuming that marriage is even recognized in the eyes of Westeros since polygamy is very much looked down upon...) ?

I am looking for theories or speculation based on the books, the TV show or historical events that end up with the people of Westeros generally accepting that Jon Snow is either:

1. 
   

   Still a bastard of different heritage, or

   
   


2. 
   

   A legitimate Targaryan, and trueborn heir of Rhaegar Targaryan.

   
   

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Edit to explain first dupe:

I am not asking who knows, I am asking: how do you prove it in the eyes of everyone in the 7 kingdoms?

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Edit based on S7E05:

In the scene where Sam decides he's had it with the silly Citadel, Gilly reads an old record of Rhaegar Targaryan receiving an annulment of his marriage to Elia Martell, and of his remarriage to... Lyanna Stark?

Assuming this is true, and assuming Jon Snow is indeed their offspring, then he is indeed a true-born Targaryan. Now as to how to prove it, obviously this record would come in handy when attempting to do so...

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Edit to explain second dupe:

This is very similar to Is there any tangible proof of Jon Snow's lineage?. However, I am asking how it would be proved, either with tangible evidence or by proclamation or by some other strategy I hadn't thought of, in the eyes of Westeros. In other words, with proof, or no proof, how are the people (nobles+commoners) of Westeros to be convinced of this truth? I accept that this question is related, and should certainly be linked, but I don't think it is a duplicate.

Answer

As of Season 7, Episode 5, Eastwatch the ways of being proven are adding up (as you have duly noted yourself).

1. Howland Reed, being present at the Tower of Joy has first account of what transpired. The proof is that he is well respected noble lord and has no reason to lie about such things as he was on the opposing side of the Targaryen's during the rebellion.


2. Brandon Stark has "green-seer" vision in which he has ascertained this information. The proof is a bit harder here as magic is going to be looked on with a certain degree of skepticism. However, if Bran can prove he knows other things (e.g. "Chaos is a ladder) that he would no other way of knowing, people may tend to believe him.


3. High Septon Maynard issued an annulment of Prince Rhaegar's marriage. While we do not 100% know, yet, that this means he married Lyanna Stark, all arrows point in the right direction. This is crucial as it wasn't just any old Septon, but The High Septon! This lends the religious authority to have Jon be a legitimate Targaryen. Even though the Sept of Baelor was destroyed, the Faith of the Seven is still the dominate religion in Westeros. Cersei may not want to hear it, but many lords and common folk will take the record seriously.


4. Jon being "dragon sensitive". I think we saw some this in S7E5 as well, when Daenerys returns to Dragonstone and Jon reaches out to Drogon. This is more speculative than the rest at this point, but seems like it will pan out. It is generally accepted that you need the blood of Old Valyria (i.e. Targaryen) to have this affinity with dragons. Update for S8E1.... Jon boi being flying now.

All of this combined would likely convince enough of the lords of Westeros to ultimately believe in Jon's claim... and if not Jon's side has dragons...